MinT - Best Known (233−89, 233, s)-Nets in Base 3

Best Known (233−89, 233, s)-Nets in Base 3

(233−89, 233, 156)-Net over F₃ — Constructive and digital

Digital (144, 233, 156)-net over F₃, using

11 times m-reduction [i] based on digital (144, 244, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 122, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(233−89, 233, 243)-Net over F₃ — Digital

Digital (144, 233, 243)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(233−89, 233, 2785)-Net in Base 3 — Upper bound on s

There is no (144, 233, 2786)-net in base 3, because

1 times m-reduction [i] would yield (144, 232, 2786)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 494 398953 855857 433293 897003 546450 452243 039716 362910 219825 645804 458391 224718 288174 412316 747806 880320 106370 526905 > 3²³² [i]